On distribution’s constant slowly varying component

In the present report it is proved that for a priori given numbers $\rho\in(1,+\infty)$ and $L\in R^+=(0, +\infty)$ there is a distribution $\big\{p_n\big\}_1^{\infty}$ with the following properties: $\big\{p_n\big\}_1^{\infty}$ varies regularly as $n\to +\infty$ with exponent $(-\rho)$, exhibits the constant slowly varying component $L$, and $\big\{\lg p_n\big\}_1^{\infty}$ is downward convex.